On a generalization of decomposable maps on C*-algebras

We propose the notion of countable decomposability of maps on C*-algebras: a bounded linear map $\varphi : \mathscr{A}\to B(\mathcal{H})$, where $\mathscr{A}$ is a C*-algebra and $\mathcal{H}$ a Hilbert space, will be called countably decomposable if it admits a representation $\varphi = \sum_{k=1}^{\infty} \psi_k \circ \phi_k$ for completely positive maps $\psi_k : \mathscr{A}\to B(\mathcal{H})$ and bounded *-maps $\phi_k : \mathscr{A}\to\mathscr{A}$. A characterization of countable decomposability is given in certain cases with various assumptions imposed on maps $\phi_k$. Our findings provide extensions of a classical result of St{\o}rmer from Proc. Amer. Math. Soc. 86 (1982), 402-404, originally formulated for decomposable positive maps.